A first group of technical preliminaries relate to semicontinuity
and convexity properties of the merit functions \(\nu \)  and \(\nu _{+K}\) .
Recall that, according to [1]}, given a closed, convex cone \(C\) ,
a mapping \(g:\mathbb {X}\longrightarrow \mathbb {Y}\)  between Banach spaces
is said to be \(C\) -l.s.c. (resp. \(C\) -u.s.c.) at \(x_0\in \mathbb {X}\)  if for any
neighbourhood \(V\)  of \(g(x_0)\)  there exists a neighbourhood \(U\)  of
\(x_0\)  in \(\mathbb {X}\)  such that
\(g(x)\in V+C, \qquad \hbox{(resp. $g(x)\in V-C$)}\qquad \forall x\in U.\) 
